INTRODUCTION
In the contradiction lies the hope
Bertolt Brecht
Performing
the call
In
my original international call for papers for a special
edition of Educational Insights, I wrote the following:
The
title, Performing the Sign: mathematics, democracy
and the arts, is intended to incite a range of responses to
themes created by the constitutive relationship between mathematics, democracy and the arts, broadly defined. The multiple significations and
interpretations produced through the performance of the
words in the title in relationship to each other and
to discourses in the social domain open up and signal
possibilities of engagement with critical theoretical
ideas and innovative textual productions.
Our
intention is to publish articles that readers may find
evocative and provocative. It is hoped that these articles
will elicit novel perspectives, thereby contributing
to understandings within mathematics education, curriculum
studies, cultural studies, and/or arts-based or infused
inquiry, teaching and learning. The articles may employ
ideas and tenets from semiotic theory, performance theory,
critical theory, contemporary socio-cultural and political
theories, to name a few, and may be interdisciplinary
or integrated in approach. Possible focus areas and questions
could be:
How
might mathematics, with its “high symbolic content,”
act as a discourse of power, oppression and/or possibility,
and what might the role of the arts be(come) in providing
a democratic face to mathematics? What assumptions might
be troubled about mathematics, its relationship, or not,
to democracy, and the alienation of the arts through
the dominance of the mathematical sciences in schools
and society? How might technology perpetuate or disrupt
the status quo on this?
What
role might the arts have in contesting and addressing
some of the undemocratic and hegemonic mathematics education
practices in schools, or might some of the advocacies
of the arts in this role be simplistic? What significations
and performances of the ‘body’ and/or of ‘place’ have
metaphorical or literal significance for mathematics,
science and arts practices in various contexts?
How
might mathematics act as an over-determined signifier
of how we have chosen to construct and live in society?
How might that shape the way we value the arts, and how
does this relate to our understandings or significations
of democracy in Western society, or the global context,
for example?
What
historical or contemporary role does religion or spirituality
play in the multiple relationships between mathematics,
democracy and the arts? How might ecological or poetic
ways of knowing provide insights into some of these relationships?
What critical philosophical issues arise in the conceptual
landscape of these various rationalities, particularly
with regard to education and educational research?
These
are only a very few suggestions intended to invite dialogue
and provide some ideas that may be addressed in the articles,
but there is scope to attend to many others. The editors
at Educational Insights welcome novel articles that are thought-provoking,
critical, and cutting-edge in theoretical approach and
design.
I could
never have imagined what a wonderful array of fascinating,
unique, innovative, thought-provoking and timely contributions
would show up in my computer’s inbox when I sent out
the call for the special issue. I had kept the call purposefully
open-ended. I was more than delighted when I realized
the call had been taken up with enthusiasm and passionate
commitment to an enlarged and complicated conversation.
It was a conversation that delved deeply and creatively
into the difficult interplay between mathematics, democracy
and the arts. Ten articles in particular stood out. They
did so in the brilliance with which they took on the
themes and took up the challenges suggested in my call,
and went much further. The arguments, ideas, associations
and perspectives were diverse and refreshing, keeping
in play the complexities, criticalities and multiplicities
of thinking and feeling that such a focus could evoke.
The challenge set in motion an intertextual discursive
(dis)play of ideas and performances that were open, rich,
informative and demanding of attention.
By opening
the curtains to the performance of possible, albeit fluid
and unstable, semiotic interconnections between mathematics,
democracy, and the arts, however these terms were to
be constituted and interpreted, was to invite the audience
into the conversation about script(s), direction of multiple
plots, and the production of a play that could have a
myriad of endings, or more likely, no ending at all!
This did not mean an ‘anything goes’ approach, but it
did not foreclose on any possibilities while engaging
with critical issues and debates that influence lives,
impact society, and speak to lived experiences.
Bakhtin
(1981) reminds us that words carry the remnants of meanings
from the places they have touched, like fragments of
old cloth. And, where the wind has taken them, they carry
with them the grit of the places where they have diasporically
lived and the discursive ways they have been performed
in context. Fragments of these meanings were evoked like
gossamer ghosts in their playful intertextual dances
between mathematics, democracy and the arts / aesthetics;
the traces and threads of which formed reticular bonds
of meaning with each other. Hermeneutic, phenomenological,
ecological, philosophical, political threads… Threads
of threads. Other dances broke the fragile filaments,
but never for long as the weaving and dancing began again.
While
somewhat playful at times, the issues were embraced with
a seriousness and passion for (re)imagining something
bolder, better, more worthy, more wholesome, more possible
for education in particular, and society in general.
For we cannot be naïve about the fact that to perform
the words, “mathematics, democracy, and the arts” together,
to gather them together in a phrase, is to evoke critical
issues of power and knowledge that have helped create
a vision of a world in which is embedded great epistemic
injustice, vast inequalities, and serious issues of oppression,
as much as it is to pose possibilities for viable, creative
alternatives. There is no neutral or objective place
to hide here. These discursive performances delve into
critical ontological and epistemological tensions, so
that we first need to understand how each of ‘mathematics,’
‘democracy’ and ‘the arts’ came/is coming to be as such,
to exist, to be understood.
While
it is the topic of this special issue, the often fractured,
often mutualistic associations between mathematics, democracy
and the arts are not arbitrary. As each has interpretive
impact on how we see the world, so the discursive relationships
between them are invested in agency and position. On
the one hand, myths, stereotypes, homogeneities, reifications,
binaries, hierarchies, divisions, antagonisms, pathologies,
oppressions, are inevitably (re)produced, altering identities,
changing subject positions and foreclosing on particular
‘realities.’ On the other hand, via shifting contexts,
hope, innovation and possibility can arise. Critically
and studiously focusing on these discursive relationships,
tensions and interconnections helps prize open common
assumptions, reinvent new ways of engaging, foster new
meanings, perform new interpretations, and address troubling
concerns that negatively affect lives and limit possibilities.
Studying
these relationships from multiple positions and “posthuman”
perspectives helps us understand how they come to constitute
and are constituted by the world, even a world that might
be conceived of as pluralistic or cosmopolitan. More
important, such a focus on the performative play between
mathematics, democracy and the arts draws attention to
how the relationships, tensions and the significations
between them might serve as moments of articulation that
might aid in (re)imagining other ways of being and coming
to know. However, it is critical to keep questions in
constant play about how these are to be affirmed and
affected, as well as in the name of what and whose ethical
responsibilities and commitments.
Drawing
on Noel Gough’s (2006, 2009) concept of “rhizosemiosis,”
a neologism derived from the geophilosophy of Gilles
Deleuze and Félix Guattari, we might begin to understand
how the tensions and attractions, the repulsions and
ramifications of the shifting reticular relationships
between mathematics, democracy and the arts operate to
try to settle and define the world, or disrupt dominant
conceptions of/within it, while simultaneously keeping
in play their complexity. It is in this critical rhizomatic
complexity and the complicatedness of their relationships
that a frond of hope may be unfurled, opening up other
versions of a vision of the world, and offering new potentially
liberating engagements and encounters.
Invitation
into the texts
It
is my honour to invite you into the worlds of each of
the contributing authors as they delineate and perform
their ideas on the intersections and relationships between
mathematics, democracy and the arts. They do so in novel
ways in both form and design. In each case, and with
differing emphasis and from different perspectives, they
do so in ways that are intentionally evocative and provocative.
In each case, experiencing their texts offers delight
and insight.
As
an enticement, I provide hints of the fascinating paths
you might walk along with the authors as you journey
into their texts and share in their narratives:
Nathalie
Sinclair and David Pimm in their article, The Many
and the Few: Mathematics, Democracy and the Aesthetic,
invite you to consider the aesthetics in mathematics,
mathematicians’ practices, and in the classroom, as a
question of taste and judgment. Drawing on the political
scientist Josia Ober’s exploration of the etymologies
of the term ‘democracy,’ they provide a working definition
of it as ‘the capacity to do things,’ which certainly
helps to facilitate their arguments. The weakest connections,
in general, are those made between mathematics in particular,
and democracy. These arguments are not well-oiled in
the mathematics education arena, although they are beginning
to become more of a focus of attention. Engagement with
democracy in this field has been somewhat superficial
at best, but deeper engagement is beginning to emerge.
This is somewhat ironic for a subject area that has proved
to be amongst the most divisive on the school curriculum,
and whose social impact on societal structures and individual
lives is enormous. It is therefore a great pleasure to
follow David Pimm and Nathalie Sinclair’s careful attention
to what they mean by democracy, which gives their arguments
definite clarity and impact. It is an aesthetically beautiful
and masterly written piece! In the earlier part of their
essay, they pose the question: “ … in what sense mathematics
is or should be seen as a ‘public good’ ?’’ It is a question
that could keep us pondering for a long while.
Steven
Khan, in Performing Oneself Differently: A Mathaesthethician’s
Responsibility, weaves personal narrative of an assault by
a group of youths to give an account of the ethics of
encounter with the Other. He asks: “What
if we began thinking about mathematics education starting
with ethics?” And explores this question in the relationships
between mathematics and art/aesthetics while probing
the possibilities for creating more responsible global
societies. He asks whether we might do mathematics
education differently, by using what we might call a
“mathemaesthethic disposition,” one that might provide
opportunities for the transformation and healing of the
glocal pathologies in which mathematics and mathematics
education is implicated. Just as David Pimm and
Nathalie Sinclair ask if mathematics should be a ‘public
good,’ so similarly does Steven Khan ask what the purposes
of mathematics and mathematics education are and, drawing
on the work of Emmanuel Levinas here, whether they might
speak to an ethical responsibility to the Other that
goes deeper and might be more personally meaningful.
Be prepared to enjoy the literary beauty and feeling
in Steven’s writing here! Stirring stuff indeed!
Iben
Maj Christiansen, in Using Art in Teaching Philosophy
of Mathematics and why it has Nothing and Everything
to do with Democracy, engages with a personal narrative of teaching a philosophy
of mathematics course at a South African university.
By bringing in an art activity with which the students
engage, Iben explores the emotions of students in gauging
their perceptions of mathematics. She uses as a working
idea of democracy, ‘perspective pluralism,’ to ask questions
about student perceptions of mathematics, their personal
philosophy of mathematics, and by extrapolation, their
own personal philosophy of life. Her thinking takes on
the broader political context of post-apartheid South
Africa, asking difficult questions about the purposes
of teaching mathematics, mathematics philosophy, and
teaching in general. In this context, the importance
of a democracy that invites and gives permission to ‘perspective
pluralism’ can, for Iben, never be overrated. In the
end, she asks: “Did we contribute to the rainbow nation? I do not know. I only hope.”
A captivating, moving piece!
In
my own article, Genu(re)flections: mathematics, democracy
and the arts, I explore how mathematics, as a discourse
of power, is implicated in neoliberal economic globalization,
and in this sense its “intrinsic dissonance with democracy,”
(Skovsmose and Valero’s, 2001). I delve into the performance
of democracy through the performance of rhizosemiotic
(Gough, 2006, 2009) relationships between and within
mathematics, democracy and the arts. Drawing on Louise
Richardson’s (2009) quote, and applying it more broadly
to education and society, I ask whether our current dominant
world view is not a form of human stupidity for forgetting
what we were trying to do—to generate understandings
of where we have been, where we are, where we might go,
and what it might mean to be human. A critical forum
for such an exploration can be provided by the Arts (including
Social Sciences & Humanities). This is in consideration
of the many manifestations of science in the social domain.
Often caught up in the pull of technoscientific/industrial
utilitarianism operationalized through economic globalization,
much science in this structural condition more often
than not contributes to failed democracy. The mass devotional
genuflecting to mathematics, technology and the sciences’
promise to foster economic development, societal ‘advancement’
and to ‘save the world’ comes at a price. The contradictions,
ruptures, falsities of logic and failed promises demand
an examination of the dominant vision, the meta-narratives,
the ideologies to which we pay homage. Inverting the
power relations, a more reflexive science is advocated.
The Arts promise a supportive critical forum for this
work and an opportunity to reclaim the public, the political,
and the human, leached by neoliberalism. It is here that
we may thoughtfully and creatively generate pluralist
(posthuman) understandings of what it might mean to be
human; human understandings that include witnessing and
remembering, despite the complicatedness and tenuousness
of such endeavours. The article ends with a narrative
of my involvement with a collaborative integrated Visual
Art-Mathematics project, a Sierpinski installation, similar
to the one Darren and Wayne describe in their article.
It has a surprising and evocative embodied outcome. Understandings
generated from this experience speak to critical relationships
and everyday social injustice. The narrative experience
draws together the political and the epistemological
in exemplifying the need for critical public fora on
education and society.
Peter
Appelbaum, in Against
Sense &
Representation: Researchers as Undetectives, uses Cantor’s proof that the cardinality (number of elements in a set)
of the power set (set of all subsets) of a set is greater
than the original set. Cantor uses a reductio ad absurdum argument. Such an argument proves a contradiction by starting out with
the assumption that something is true that the mathematician
believes to be false and therefore tries to convince
her/his audience that it is indeed false through the
contradiction produced: a somewhat odd logic that has
been used for centuries. Peter moves from this oddity
to the fantasy and (non) fiction of a fascinating collage
of stories—ones that encompass mathematics, mathematicians,
students, politicians, representations, and communities.
They are stories in which he “finds solace in post-modern
‘undetectives,’ whose mysteries are not quite solved
in the sense of a resolution grounded in truth.” In the
(non-)end, his stories are of deception, intrigue, parody
and misrepresentation. They tell much about so many things—art,
mathematics, schools, curriculum work, life. Masterfully-written,
his article will delight and intrigue you too!
In
Susan Gerofsky’s article, Performance Mathematics
and Democracy, she deploys arguments for a pedagogy
of school mathematics based on the concept of participatory
performance art for social change. Drawing on theories
of technology and cultural / historical change, and that
of space and liminality grounded in embodiment and performance,
Susan offers examples of initiatives in developing a
pedagogy of school mathematics as democratic performance
art. Amongst other suggestions, she advocates for a move
from the visual, which has dominated mathematics pedagogy
for decades, to include more of the auditory and musical.
Her work is timely and insightful in offering an alternate
vision of school mathematics pedagogy that pulls us out
of both traditionalist and (some of the more stale) progressive
ruts. Drawing on her vast experience in media, theatre,
movement, the creative arts, and secondary school and
university teaching, Susan’s democratic vision and ethical
consciousness of sustainable living and justice-oriented
learning opens up possibilities for innovation and a
greater promise for living and learning well. I have
no doubt that her article will be thoroughly appreciated
and enjoyed!
Elizabeth
de Freitas, in Making mathematics public: Aesthetics
as the distribution of the sensible, takes
a bold step into the philosophy of Jacques Rancière and
his political reading of aesthetics as a way of rethinking
the relationship between aesthetics and mathematics.
Aesthetics regulatory role in constituting “the sensible”
in the public domain has another side to it. Art that
has troubled the rules of representation has also disturbed
what is taken to be commonly held as a shared public
reality by a particular community. Elizabeth de Freitas
takes an unusual approach. She uses this thinking to
try to make sense of the semiotic ‘written-ness’ of mathematics
and the important role of surfaces in doing mathematics.
This moves beyond the historical Euclidean space of mathematics
to the tactile textu(r)al gritty reality of writing and
doing mathematics by focusing on the sign-making forms
that operate on surfaces. As one reads her work, art
and mathematics find approximations, and one feels and
appreciates the surfaces touched—philosophical, conceptual,
and material surfaces, in the signing and significations
of mathematics and art making. Quite the cerebral-aesthetic
experience! Elizabeth de Freitas traces this thinking
in relation to historical participation in mathematics
practices and the subject positions produced. A truly
novel approach!
In
their article, On the Primordiality and Poiesis of
a Complexified Performance, Darren Stanley and Wayne Tousignant draw
on Complexity Science to capture the “patterns of patterns”
that might bring into focus understandings of the complex
phenomenological meanings that emerge from a collaborative
art–mathematics project done with university students.
The collaborators describe the design and construction
of a mathematical sculpture created for a fire sculpture
festival. The Sierpinski tetrahedron structure is full
of the joyously aesthetic principles of self-similarity
and fractal geometrical delight. They use it to narrate
and reflect back upon their own learning, conversations,
and final resulting mathematical installation in (a)light
of what is currently known about complex dynamical systems.
The experience gives rise to considering the phenomena
of the experience in terms of what may constitute democracy
and what may be understood by democratic phenomena, which
for Darren and Wayne are phenomena that are inherently
healthy learning organizations. These are ideas that
are healthily provocative! I can assure you that you
will be delighted and transfixed by Darren and Wayne’s
descriptions and images, and the novelty and aestheticism
of their arguments!
Graham
Giles offers a very unusual, densely philosophical article
suggested by the novelty of his title: Pigs, Stars,
Gods, and Alain Badiou’s Mathematical Language of Being.
It requires careful reading, but is immensely stimulating
and provocative. His objective is to offer some key elements of Badiou’s
philosophical thought as new points of departure for
the thinking of mathematics in education, and, more broadly,
in education newly of being, truth and the subject—Badiou’s
foremost preoccupations. One enters into glades and forests
filled with dim and brightly lit lanterns of ideas, concepts
and thoughts as you journey with Graham. In this article,
following from the thinking of Alain Badiou, Graham develops
some of the central ideas of Badiou’s mathematical (set
theory) “metaontology” toward a critique of the cant
of democratically authoritative concepts of the Other,
Difference and Plurality. Drawing on Badiou’s incisive
distinction among the “grand style” and “little style”
thinking of mathematics, Graham argues that the reconstitution
of philosophy via Badiou’s mathematical metaontology
opens a place to consider the “possibility of possibility
itself” and thus reconsider the ethical in education
in ways that have implications also for politics, art,
science and love. A truly magnificent, insightful read!
Patti
Pente and Gladys Sterenberg invite us into an engaging
visual, philosophical, aesthetic experience with Signs
of Zero. The
artistic is imbricated in the reading / seeing experience,
the image and the word merge. The message is the medium,
and the medium is the message. Drawing together the experience
of the artist in her community and the mathematician
in hers, they attend to the assumptions within the hierarchy
produced in the social domain that has set up the mathematics
/ art binary, and that of actual experiences of mathematics
and art as a contradiction. They note that, “Throughout
history, mathematicians (like artists) have […] been
part of the community. Interpretations of mathematicians
or artists as contemporary social critics, and as participants
in the creation of social imaginaries that can inform
public spheres, firmly place them within community.”
They play on spaces and concepts of “zero,” historical,
conceptual and paradoxical, to debunk myths about mathematics
and the arts, and they offer us an engaging and deeply
aesthetic and imaginal experience of their experiences
along the way. A delightful, evocative and sensate experience!
Margaret
Walshaw’s, The Performance of Self in the Art of Research, offers
insight into the tensions, mental-emotional negotiations
and experiences in doing and living research. Her article
focuses on the performance of the self as researcher,
both within the ‘data gathering process’ and the construction
of research reports. Drawing on understandings of ethical
practice, it grapples with what it is that structures
a democratic narrative experience. There is a dual objective
here: one to better understand the workings of subjectivity
and the intersubjective, the other to scrutinize the
researcher’s ‘self.’ This necessitates an authentic delving
into uncomfortable places—places beyond sociological
constructs of subjectivity, discourse and power, although
it includes a careful study of these as well, but to
human emotions, desires and aspirations, to uncover the
layers of self and selves to explore the unconscious
in the way it interferes with and informs the performance
of the art of research. Margaret Walshaw provides a personal
account of doing mathematics education research with
young women in schools in Australia to explicate her
arguments and ideas, using the writing to explore the
unconscious self. While making an important contribution
to reflexive inquiry, it is a careful, thoughtful and
not-to-be-missed reading!
The
performance begins
There
is a hush, anticipation in the air, the curtain sweeps
aside, the first note has sounded … the performance has
begun! ….
I wish
to thank all the authors for their remarkable contributions,
for the discussions we have engaged in and the humanity
we have shared. I wish to thank them for the excellence
of their work and of the passion with which they have
embraced the objectives of the call to this special issue.
In particular, I would like to thank Lynn Fels for her
ethical presence, remarkable support and unbending humanity.
I wish to thank Graham Giles for much the same: his editorial
and personal support, insight and humanity. I also thank
Martin Elliott and Marshall Fels Elliott for their creative
genius, wonderful support and dedicated efforts, as well
as Michael Boyce for his incredible and supportive editorial
work. Without the remarkable support of all these people,
this special issue would not have been possible.
Dedication
I dedicate
this special issue to all those who have suffered under
the tyranny and ignorance of our forgetting what we were
trying to do—to “generate (pluralist) understanding
of where we have been, where we are, where we might go,
and what it means to be human” (Richardson, 2009).
References
Bakhtin,
M.M. (1981). The Dialogical Imagination. M.
Holquist (Ed.), Trans. By C. Emerson. Austin, U.S.A:
University of Texas Press.
Gough,
Noel. (2006). Rhizosemiotic play and the generativity
of fiction. Complicity: An International Journal of
Complexity and Education, 4(1), 119-124. http://www.complexityandeducation.ualberta.ca/COMPLICITY4/documents/Complicity_41l_Gough.pdf
Gough,
N. (2009). Becoming Transnational: Rhizosemiosis, Complicated
Conversations, and Curriculum Inquiry. In M. McKenzie,
P. Hart, H. Bai, & B. Jickling (Eds.), Fields
of Green: Restorying Culture, Environment, and Education, (67-83). Cresskill NJ: Hampton Press.
Richardson,
L. (2009). Vice-Chancellor Address at the November 2009
Graduation Ceremony of the University of St. Andrews:
http://www.st-andrews.ac.uk/news/archive/2009/Title,44179,en.html
Skovsmose,
O & Valero, P (2001). Breaking Political Neutrality:
The Critical Engagement of Mathematics Education with
Democracy. In
B. Atweh, H. Forgasz & B. Nebres (Eds.), Sociocultural
Research on Mathematics Education: An International Perspective
(37-55). New Jersey: Lawrence Erlbaum Associates.
